The main hyperparameters used for constructing the descriptors and kernel
functions [1]}, [2]} are as follows. The cutoff radius
used to represent the local environment of each atom was set to 6 Å, while
the width of the Gaussian functions used to broaden the atomic distributions was
0.4 Å. These parameters were the same for the two- and three-body descriptors.
To expand the atomic distributions, up to 15 spherical Bessel functions (for
the angular quantum number \(l=0\) ) and Legendre polynomials of order \(l\)  up to
\(l_\textup {max}=4\)  were employed, resulting in 2048 descriptors for
the Zr and O atoms. The hyperparameter for the dot-product kernel
was set to the standard value \(\zeta =4\) . The automatic sparsification of the
three-body descriptors based on the CUR algorithm was applied [2]},
allowing us to halve the number of descriptors needed without affecting the
accuracy of the MLFF. As described previously, the MLFF can be further
tuned by reweighing the energies, forces and stresses differently when setting
up the linear regression problem. Here we found that weighing the energy equations
10 times more strongly reduces the errors in the energy by about 0.5 meV per atom
on average, while increasing the errors in the predicted forces and stresses
only marginally. Finally, both BLR and SVD were adopted to determine the
regression coefficients.
